^{*}

Edited by: Jingming Hou, Xi'an University of Technology, China

Reviewed by: Alexander Maria Rogier Bakker, Directorate-General for Public Works and Water Management, Netherlands; Jianmin Zhang, Sichuan University, China

This article was submitted to Hydrosphere, a section of the journal Frontiers in Earth Science

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The Discrete Boltzmann Equation (DBE) is a versatile simulation method, consisting of linear advection equations, which can be applied to the Shallow Water Equations (SWE). The aim of this study is to assess the accuracy of the DBE to simulate dam-break of shallow water flows in presence of obstacles. Dam-break flows in the presence of obstacles can be regarded as simplified models for floods in urban areas. In this study, three cases of dam-break flows in the presence of obstacles are considered: two with an isolated obstacle and the third in presence of an idealized city. A comparison between DBE and benchmark results shows a fair agreement, confirming the validity of the DBE as simulation method for the SWE.

Flooding is considered one of the most significant hazards society is facing. Huge losses in terms of human lives, damages to buildings, houses, and civil infrastructures are caused every year by hurricanes and flooding. Moreover, the impact of flooding in the future is expected to become increasingly important due to deforestation and depopulation of rural zones (Sanders and Schubert,

Urban flooding are particularly fearsome events, considering that more than half of the world's population lives in urban areas and that the urban population is continuously increasing (Song et al.,

The main mathematical model for flood simulation is based on the two-dimensional Shallow Water Equations (SWE), which is obtained by averaging mass and momentum balance equations along the vertical direction under the classical assumption of hydrostatic pressure distribution (Valiani and Begnudelli,

An alternative option is represented by the Smoothed Particle Hydrodynamics (SPH) formulation of the SWE (Chang et al.,

Solution methods for the SWE have also been developed in the Lattice Boltzmann Equation (LBE) framework (Zhou,

The aim of this paper is to assess the ability of the DBE in simulating dam-break flows impacting on obstacles as a preliminary step for the simulation of urban flooding events on real topography. Indeed dam-break flows impacting on obstacles can be considered as simplified models of urban flooding, and the scientific literature abounds with experiments and numerical simulations of dam-break flows impacting on obstacles. In this paper, the experimental and numerical dam-break flows of Kleefsman et al. (

In these works, water depth and velocity are measured at given measurement points and/or are numerically calculated by means of previously validated finite volume-based numerical solvers either applied to the SWE (Soares-Frazão and Zech,

The structure of the paper is as follows: first, the mathematical model is briefly presented; second, the considered benchmarks are described; third, results are reported and commented on; and fourth, conclusions are drawn.

The SWE, obtained by averaging mass and momentum balance equations along the vertical direction under the classical assumption of hydrostatic pressure distribution (Valiani and Begnudelli,

where _{m} is the Manning friction coefficient.

The first term at right hand side of the second and third Equation (1) accounts for the bottom slope and is calculated as in Valiani and Begnudelli (_{f}+_{f} the bottom elevation. This formulation of the bottom slope term ensures the correct balance between flux gradients and source terms (Valiani and Begnudelli,

The DBE is given by (La Rocca et al.,

where _{k} is the probability density function relative to the _{xk}, _{yk}. The particle velocity set as the ^{*} is the dimensional relaxation time,

The DBE (2) is equivalent to the SWE (1) in the sense that the water depth _{k}

coincide with the water depth

It is worth observing that the DBE (2) that consists of a set of uncoupled, linear advection equations with constant advection velocity; this is the most appealing aspect of the DBE, when compared with the highly non-linear SWE, and it is an advantage

The DBE (2) is discretized by means of an explicit first order approximation of the time derivative and a first order upwind approximation of the space derivative (La Rocca et al., _{max} × Δ^{−1}), _{max}, Δ^{*}/Δ

This case has been considered in Kleefsman et al. (_{1}, _{2}, _{3}, _{4}) and four pressure probes (_{1}−_{4}) (Kees et al.,

In _{1}, _{2}, _{3}, _{4}) and pressure (_{3}) probes is also reported in

Test case I. Experimental and numerical free surface profiles at _{1}, _{2}, _{3}, _{4}; dot on the obstacle: position of the

In _{1}, _{2}, _{3}, _{4} is plotted. Dots represent the experimental results obtained at MARIN, while the solid line represents the DBE numerical results. The latter are those obtained with the finest grid (800 × 280). The space step Δ^{−3} ^{−4})_{3} is shown in

Test case I. Time history of the water height at probes _{1}, _{2}, _{3}, _{4}. Solid lines: DBE numerical results. Dots: Experimental results.

Test case I. Time history of the pressure at probe _{3}. Solid line: DBE numerical results. Dots: Experimental results.

This case has been considered in Soares-Frazão and Zech (

Finally, the Manning coefficient is set to

In _{1}, _{3}, _{4}, _{6}. Solid lines represent experimental values, while dotted and dashed lines represent the DBE numerical values obtained with a coarse (600 × 67) and a fine (1500 × 165) grid, respectively. The difference between the DBE numerical results obtained with the two grids is not meaningful, mainly at measurement point _{6}. The differences with the experimental results observed at measurement points _{1}, _{3}, _{4}, _{6} are ascribed to the fact that there the vertical motions are not negligible and that the shallow water model loses consistency. Nevertheless, the DBE numerical results grab the main features of the flow, and the general agreement with experiments can be considered fairly good. The DBE numerical results are in satisfying agreement with the numerical results of Noël et al. (

Test case II. Experimental and numerical free surface time histories. Solid line: experimental measurements from Soares-Frazão and Zech (

Test case II. Free surface at

This case is reported in Soares-Frazão and Zech (

A gate (width 1

In

Test case III. Experimental dam break flow through an idealized city of Soares-Frazão and Zech (

Test case III. Experimental dam break flow through an idealized city of Soares-Frazão and Zech (

Test case III. Computed water depth map. Darker color for larger depth values. Top panels

In particular, at _{H} = 3.95 _{E} = 4_{EN} = 0.7 _{E} = 4_{ES} = −0.7 _{H}, _{E}, _{EN}, _{ES} coincide in

At _{H} = 3.4 _{E} = 3.3_{EN} = 1.0 _{E} = 3.3_{ES} = −1.0 _{H}, _{E}, _{EN}, _{ES} coincide in

The multispeed Discrete Boltzmann Equation (DBE) is a method recently developed to simulate transcritical shallow water flows, and the main advantage is the simplicity of the mathematical model: a set of linear, uncoupled, purely advective equations. In this paper, the DBE has been assessed as solution method for the Shallow Water Equation (SWE) when dam-break flows impacting on obstacles are considered. These flows are strongly unsteady and present frequent sub-supercritical and super-subcritical (hydraulic jumps) transitions.

The consideration of dam-break flows impacting on obstacles is important as they represent simplified models of urban flooding, thus constituting a benchmark for numerical methods, thanks to the wide variety of reference numerical and experimental data published in literature.

In this paper, three dam-break flows impacting obstacles have been considered: two on an isolated obstacle and one on an idealized city that has realized by means of a regular array of parallelepipeds.

The experimental configurations were realized within tanks with a smooth, though sloping, bottom. The latter has been treated in order to account for the correct balance between flux gradients and source terms.

The comparison between DBE and reference results (both numerical and experimental) shows that the DBE can be considered as a reliable solution method for dam break flows impacting on obstacles, even in the presence of complex flow configurations, such as e.g., the case of the idealized city. Discrepancies between DBE and reference results can be attributed both to the use of coarse grids and to the accuracy of the numerical method and to the occurrence of violent vertical motions, which undermine the consistency of the shallow water model. It is worth observing that, for what regards the specific performance of the proposed models in urban areas, there are no restrictions for the DBE on the attainable grid resolutions. Any grid convergence rate can be achieved by just adopting a higher order numerical scheme. Future works will be addressed to the consideration of realistic topography as well as to the implementation of more efficient numerical methods, such as e.g., the finite volume method with unstructured, adaptive grids.

The datasets generated for this study are available on request to the corresponding author.

ML has formulated the Discrete Boltzmann Equation. PP performed the numerical simulations. SM analyzed the results and prepared graphics. All the authors have discussed together about the outcomes and their interpretation. All authors contributed to the article and approved the submitted version.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Valuable discussions with Dr. A. Montessori are kindly acknowledged.

The

being _{xk}, _{yk} the cartesian components of the vector _{k} along _{k} are in turn defined:

The particle velocity vectors can be grouped into two subsets, hereinafter referred to as shells, based on their magnitude _{0}, _{0}, _{0} a velocity scale representative of the surface waves celerity. _{0} have been set to _{0} being the initial water height. The total number of velocity vectors _{T}+1 must be higher than the minimum value 13, corresponding to _{T} = 12, below which the simulation becomes unstable.

A polynomial form is assumed for the equilibrium probability density function

where the _{k}, ..., _{k} have to be determined by matching the discrete hydrodynamic moments with those obtained from the shallow water Maxwellian equilibrium distribution function

The following expressions are obtained: for

for

for

ϕ, β being defined by: